categorical h^-tensor productcategorical h^-tensor product by john daunso abstract. if a and b are...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 166, April 1972

CATEGORICAL H^-TENSOR PRODUCT

BY

JOHN DAUNSO

Abstract. If A and B are von Neumann algebras and A ® B denotes their cate-

gorical C*-tensor product with the universal property, then the von Neumann tensor

product A V B of A and B is defined as

A V B = (A~® B)**jJ,

where J <^(A ® B)** is an appropriate ideal. It has the universal property.

1. Introduction. Consider the category C* of C*-algebras and the category

W* of von Neumann algebras. There is a functor E: C* -*• W* that assigns to a

C*-algebra A its universal enveloping algebra EA = A**, or double dual. A C*-

algebra A is a von Neumann (or W*) algebra if and only if A is the dual A = A* of

some unique Banach space A* called the predual. The preduals form a category

33 of Banach spaces that is anti-isomorphic to W* under the contravariant functor

P: W* -> 93 given by PA = Ajf.

An example at the end (see Example 6.3) shows that the usual tensor product on

W* does not have the universal property. One of the objectives here is to construct a

new tensor product "_V_" on W* that has the universal property. The construction

is carried out without representing the algebras as operators on some (nonunique)

Hilbert spaces, but rather more conceptually by first constructing a product "_A_"

on 23 and then transferring it as "_V_" to the dual category W* by use of the

functors P and E. A tensor product A eg) B of C*-algebras is said to have the

universal property provided whenever a: A -h>- D and ß: B -+ D are C*-maps with

pointwise commuting images in D, then there is a unique C*-map (a, ß) : A ® B

-► D giving the usual commutative diagram. It has long been known that the

subcategory C*1<=C* of C*-algebras with identity and identity preserving mor-

phisms has a tensor product. In this setting it will be shown that

(C*l, ®) -£* (W*, V) -^ (33, A)

are functors of multiplicative categories, where the morphisms in C*l need not

preserve identities.

Consider two von Neumann algebras A = A"<^£f(H) and B=B"<=-Sf(K), where

Received by the editors November 19, 1970 and, in revised form, June 14, 1971.

AMS 1970 subject classifications. Primary 46L10, 46M05; Secondary 46L05, 46M15.

Key words and phrases, von Neumann algebra, predual, categorical C*-tensor product,

normal, singular functionals, functors of multiplicative categories, adjoint functor, cogebras,

bigebras.

O This research was supported by the National Science Foundation Grant GP-19824.

Copyright © 1972, American Mathematical Society

439

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

440 J. DAUNS [April

A" is the double commutant of A and if(H) denotes all bounded operators on a

Hubert space 77. The usual definition of the von Neumann tensor product is

(A O B)"^-Sf(H~® K), where "-O-" will always denote the algebraic tensor product.

Only as long as A and 77 are represented on some Hubert spaces such that A = A"

and 77=77", will this definition be independent of the Hubert spaces 77 and K. Two

different proofs of this independence have been given by Misonou [9, p. 190,

Theorem 1] and Takeda [15, p. 218, Theorem 3]. The exact same tensor product

has been defined more abstractly by use of preduals by Sakai [10, p. 3.17, Theorem

2.3]. Authors treating duality of locally compact groups and Hopf H/*-algebras

have consistently used the above noncategorical tensor product. Perhaps duality

theory would assume a more elegant form if the right tensor product was used.

2. Product on the preduals.

2.1. For any C*-algebras A, B with or without identities, consider all C*-homo-

morphisms a: A -> D and ß: B -> D with pointwise commuting images into some

C*-algebra D. If m: D O D -> D denotes the map defined by the multiplication

of D, let p denote the norm

pit) = sup {\\m(a O ß)(0\\ :a,ß,D}, teAO B,

on the algebraic tensor product A O 77, where the supremum is taken over all such

a, j8, and D. That p actually is a cross norm, i.e. that p(a o b)= \\a\\ \\b\\, is shown in

[6, p. 11, 3.1]. Let A Ö 77 be the p-closure of A O 77.

Only in case both A and 77 have identities, define A® B as A® B=A O B. In

case 1 £ A or 1 £ 77 or both, the functor " JgL" later will be given a different mean-

ing and definitely will not denote the above completion. If A and B have identities,

then A ® 77 is the categorical tensor product for C*-algebras, i.e. the above maps

a and ß factor through a unique homomorphism y: A ® 77 -> D. Define (a, ß) to

be (a, j8) = y. Note that D may or may not have an identity and in case it does, the

maps a, ß may or may not preserve identities.

2.2. Any von Neumann algebra A is the dual A = A% of a unique Banach space

A* called its predual. The dual of A is A* and o(A, A*) denotes the weakest or

smallest topology on A making all the functionals of A* continuous. In case A is

understood, simply write o = o(A, A%). A map between von Neumann algebras

(not necessarily a homomorphism) is normal provided it preserves least upper

bounds of arbitrary sets of positive elements. Then the a-continuous linear func-

tionals on A are precisely A*; moreover, A* can be intrinsically characterized as

exactly the normal functionals [10, p. 121, Theorem 5.1].

2.3. The norm p induces a positive extended real valued function p' on A* O 77*

defined at fe A* O B* as the supremum of the function / over the p-unit sphere of

A O Boy

p'(f) = sup {|</, ty\:teAOB, P(t) - 1},

where <_,-> will subsequently denote the dual pairing between a space and a space

of linear functionals on it, i.e. a bilinear map <-,-> : A* Q 77* x A O 77 ->- C. There is


1972] CATEGORICAL ^»-TENSOR PRODUCT 441

a least cross norm A on A O B whose dual A' is also a cross norm (see [11]). Since,

firstly, pisa cross norm and since, secondly, p ä A, it follows also that p is a cross

norm. The completion of the algebraic tensor product A* O B* in the norm p'

will be denoted as A% O Ti*. Again, if both A and B have identities, write A* ® 7?*

= AifO Bjf. Since by definition, the norm p' when extended to A* <g> 7?+ is the opera-

tor norm of AQB^C, it follows that A* g) B^(A <g> B)*. Subsequently of

main interest will be a space between the latter two.

2.4. The normed space (A ® B)* will be identified with a suitably normed space

of linear maps T: A^ B*.

For Te (AMB)* and a e A, let T(a eg) _) €7i* be defined by

r(a(g)-):Ä-^C, b->T(a®b), b e B.

Thus 7* induces linear maps

A->B*, B^A*,

a->T(a® -), b-^T(-® b),

where <7*(a eg -), b) = T(a <g> ¿>) = <a, T(- ® *>)>•

Conversely, any arbitrary linear map T: A^- B* can be regarded as a linear

functional on .4 O Ti by means of the definition 7*(2 (a O b)) = 2 <7a, Z»>. Define an

extended real valued norm 1117*111 on these linear maps T by

1117*111 = sup ||2 <Ja, b}\:Z(aOb)eAOB, P(% (a O b)\ = l|.

Note that the sup-norm of the functional T on the normed space (A O B, p) is

precisely III 7*111. Thus the condition that 7* be extendable to a continuous functional

T: A <g> 7i -> C is that III Till be finite in which case III 7*111 is simply the norm of the

extended functional. Hence (A ® 7?)* can be identified as the space of those linear

maps T: A^ B* that are finite in the norm III 7*111. In addition, the norm III 7*111 is

the norm of 7* as a linear functional on A® B (see [11, p. 45, Theorem 3.1]).

Alternatively one could consider also T: B -*■ A*.

The previous characterization of (A <g 7?)* allows us to define a product A% A Ti*

of the preduals of A and B as a space of linear maps T: A^~ B* whose range is

suitably restricted.

Definition 2.5. For von Neumann algebras A and 7?, define A* A 7?* as the

closed subspace in A* A B^(A ® 7?)* consisting of all those Te (A ® B)* such

that T: A^-B* with TA £ Ti* and also such that when T is viewed as a map

T-.B-+A*, then TB^ A*.

Alternatively, A* A Ti* is the set of all Te (A® B)* such that T(a eg -) e Ti*

and T(- eg b) e A* for all a e A, b e B.

3. Universal enveloping algebras. In addition to providing a suitable framework

for constructing the von Neumann tensor product, it will be shown, among other


442 J. DAUNS [April

things, that the universal enveloping algebra A** of an infinite dimensional W*-

algebra A contains at least two distinct isomorphic copies of A.

3.1. Consider a Banach algebra A, elements a, b, xeA;pe A*; andF, GeA**.

Then

apb e A* : (apb)(x) = p(bxa);

GpeA*:(Gp)(a) = G(pa);

PG e A* : (pG)(a) = Giap);

FG e A** : (FG)(p) = FiGp).

With this multiplication, A** becomes a Banach algebra containing A as a sub-

algebra A^A**. For any homomorphism a: A -*■ D into a Banach algebra D,

the natural map a**: A** -> D** is also a homomorphism; moreover, the restric-

tion (and corestriction) a**\A of a** to A (and D) is a**\A = a.

3.2. If, in addition, A is a C*-algebra, then A** is its universal (von Neumann)

enveloping algebra. The algebra A** together with the natural inclusion./: A -> A**

is uniquely characterized by the universal property that any C*-map y. A ->- D

of A into any von Neumann algebra D factors uniquely through y: A** -*■ 7).

If -n: D—>■ D** is the natural inclusion, then by the universal property of D**

there exists a unique normal map </>: D** -> D with <fir¡ = l, the identity on 7).

Then there is a commutative diagram

The map y** is well known to be normal while -q in general is not normal. Although

the maps y** and r¡y agree onj(A), in general they need not be equal.

Some already known facts (see [10]) are organized in the next lemma in a form

in which they will be used later.

Lemma 3.3. Suppose A is a C*-algebra with or without identity and W<^A* a

norm closed subspace invariant under translations from A. If WL is defined to be the

polar

WL = {F e A** \ FW = {0}},

then

(i) A** = A**(l— e)@ W1 where both summands are ideals and von Neumann

subalgebras; e is a central idempotent in A** and an identity for WL;


1972] CATEGORICAL H^-TENSOR PRODUCT 443

(ii) A* = W® eA* ; furthermore W=il-e)A*.

There are natural isomorphisms induced by the direct sum in (ii) and the natural

action of A** on A*:

(iii) W*Â**il-e);

(iv) ieA*)*^WL.

If D = D*, is any von Neumann algebra and A = D, W= D* above, then

(v) D^D**(l-e).

Proof, (i) It follows from [10, p. 1.74, Corollary 15.1] that WL <\ A**. From

now on the notation "<l" always will indicate a (not necessarily closed) two sided

ideal. Furthermore, W*Â**/WL. Consequently, A**/WL is a von Neumann

algebra with predual W. It follows from [10, p. 1.74, Remark 2] and [10, p. 2.3,

Theorem 1.3] that W1 is a a(A**, y4*)-closed ideal and is itself a von Neumann

algebra. Every von Neumann algebra Wl contains an identity eeW1(Â**.

Since ee W1 <¡ A**, for any FeA**, eF=e2F=eFee WL. Hence eF=eFe=Fe,

and e is a central idempotent of A**. Thus A**=A**(l — e)® WL.

(ii) For any closed subspace WÂ*, the kernel of W1 (also called the double

annihilator of W) is W, i.e.

{feA*\(W\fy = {0}}= W.

First clearly Wç kernel W1. On the other hand, consider fie kernel W1. If/^ W,

then by the Hahn-Banach theorem there exists G e A** with GW={0} but G(/)^0.

Thus Ge WL, fie kernel W1, but G(fi)¿0, a contradiction. Since Wx = A**e,

(l-e)^*£kernel WL=W. But

{0} = (W1, eA* n W) = (A**e, eA* n W) = (A**, eA* n W}.

Since eA* n Iîs annihilated by all of A**, it follows that eA* n ^={0} and that

A*= W® eA*. Thus W=(l-e)A*. It should be noted that since e is a central

idempotent, eA* = A*e.

(iii) and (iv). Since A** = W1@A**(l-e), clearly vV*Â**/WJ-zA**(l-e)

and (eA*)*Â**/(eA*y. Since (l-e)e = 0, A**(l-e)ç(eA*y. Since kernel W1

= W,a nonzero element of Wx cannot vanish on all of eA* without vanishing on

all of A* = W © eA* and thus being identically zero. Thus (eA*)1 = A**(l — e) and

(eA*)*^ W1.

(v) Now specialize the above A to a von Neumann algebra A = D = D* and

W=D* to be the predual. Then D** = D**(l -e) ® 7)¿ is an ideal direct sum of

von Neumann algebras, where D*^D**(l— e)^D.

3.4. Remark. A positive linear functional on a von Neumann algebra D is called

singular provided it annihilates arbitrarily small projections in the partial order on

D (see [10, p. 1.75, Remark 2]). The space of singular linear functionals is precisely

eD*.

3.5. Associated with each pair W, A are two intrinsic maps—the natural in-

clusion t?: A -> A** and the projection -it: A** ->- A**(l —e) onto the first factor.


444 J. DAUNS [April

Later, when necessary, the functorial nature of these maps will be indicated by r¡A

and 77,4.

The author knows of no place in the literature where either the next proposition

or corollary is proved.

Proposition 3.6. Consider any C*-algebra (with or without identity) and

any norm closed translation invariant subspace WCA*. If W separates the points

of A, i.e. W, A are nonsingularly paired, then r¡A r\ A**e = {(S} and the map

777j: A -> A**(l—e) is monic.

Proof. By definition of r¡: A-> A**, for r¡aeA, we have (r¡a, W) = (W,a}.

Thus the spaces r¡A and W are nonsingularly paired. It follows that

(A**e, W} = {0} => r]A n A**e = {0}.

Finally, since 17 is monic, since kernel ir=A**e, and r¡A n A**e={0}, it follows that

■nr¡: A ->■ A**(l-e) is monic.

Corollary 3.7. 7/i'h the above proposition A = D = D%is a von Neumann algebra

with W= 7)*, the predual, then m) is an isomorphism and there exists a normal iso-

morphism of von Neumann algebras 6 with

D -^-> D** -^* 7J>**(1 - e) —► D

and such that 8tttj=1, the identity on D. Furthermore dir = ip, where ifr: D** -> D is

the unique normal map with 1/177 = 1 on D.

Proof. By 3.6, irn is monic. It is asserted that ■mn(D) = D**(l - e). Suppose not.

Let Fe D**(l-e)\7rq(D). Since Fe D**(l-e)^D% and 7)*s7>, there necessarily

exists an element de D with <F,/> = </ cf> for all fe 7J>*. But </ d> = <r)d,f>.

Consequently, {F-r¡d, 7>*>={0} and F—nde D^ = D**e. Thus

F—qd = (F—nd)e = -(rjd)e er¡Dr\ D**e = {0}

by 3.6, a contradiction.

Since each element z e D**(l—e) is uniquely of the form z = (r¡d)(l — e) with

de D, define a C*-isomorphism 6: D**(l —e)-^Dhy 6z = d. However, any iso-

morphism of von Neumann algebras is automatically normal. Thus there are maps

777} . 6D—^D**(\-e)—>D

with 7777 monic, with 6 an isomorphism and with dtrr] = 1. Since both ip and 8n are

normal maps with Ô777; = i/itj = 1 on D, by the uniqueness of the map D** -> D for

the universal enveloping algebra of D, it follows that 677=1/1.

Remark 3.8. If D = D* is an infinite dimensional H/*-algebra, then D*=£D*,

and hence 7»^; = 7J»**e^{0}. Then D^-nD^D** and also D^D**(l-e)^D**

are two distinct isomorphic copies of D inside D**. They are distinct because D**


1972] CATEGORICAL W*-TENSOR PRODUCT 445

is the a-closure of r¡D and D**(l -e) is a von Neumann subalgebra of D**, i.e.

since D**(l-e) is tr-closed, r¡D^D**il-e) would imply that the a-closure of r¡D

is also in D**(l—e).

4. The von Neumann tensor product. The tensor product defined below is

different from the usual one in [10]; the one in [10] is also equivalent to that

considered in [9] and [14].

4.1. Consider von Neumann algebras A and 77. Since A* and 77* are translation

invariant, A* A 77* is invariant under translations from A O 77 and hence also

A® B. Clearly, A* ® 77*£.4* A 77*.

Definition 4.2. For any von Neumann algebras A, B define their tensor product

A V 77 as the von Neumann algebra

AV B=(A®B)**/(A*&B*y.

(Since (A* A 77*)* ^4 V 77, we have (A V 77)* = ,4* A 77*.)

In the next lemma as well as throughout this section it is assumed that homo-

morphisms preserve identities.

Lemma 4.3. Suppose a : A -» D and ß: B -> D are normal identity preserving maps

of three von Neumann algebras A, 77, D with pointwise commuting images. Consider

the resulting map y: A ® 77 -► D on the C*-tensor product and the induced maps

y*:D*-+(A® 77)*, «*: £>* -> A*, /3*: 7)* -> 77*. Then y*7)*<=,4* A 77*.

Proof. Suppose p e Z>*. Then y*p : A ® 77 ->• C. It suffices to show that for any

fixed a e A, the linear functional y*p(a <g> _) : 77 -► C is in 77*. The translate pa(a) of

p is also in D*. Thus y*p(a (g> -) = [pa(a)]ß, where

77 l^D^lc

is a composition of the two normal maps ß and pa(a). Hence their composite

[pa(a)]ß is also a normal map and y*p(a (8> -) e 5*. The argument for showing that

y*p(- (8) b)e Ai, is entirely parallel and is omitted.

Lemma 4.4. Assume the hypotheses of Lemma 4.3 and denote by y the unique

extension of y = (a,ß) to the enveloping algebra (A (g) 77)**. Suppose j: A® B

-> (A <g) 77)** andr¡: D -> D** are the natural inclusions while it and 9 are the maps

of 3.7. 772?« y**j=-qy, y=diry**, and there are commutative diagrams

A®B > (A®B)**

£)**

fr-D**il-e)^n


446 J. DAUNS [April

Proof. The first diagram is obtained by combining the commutative diagrams

of A eg) 7? and (A ® 7i)**. Since by 3.2, -qy=y**j, we have B-m]y= 6try**j. But use of

3.7, i.e. the fact that #7777 = 1, shows that y=0-ny**j. Thus the second diagram also

commutes.

Proposition 4.5. Under the hypotheses of Lemma 4.3, there is a normal map q and

a commutative diagram

(A V B)%

■v

Proof. Quite generally, even for Banach spaces

y*(7)#) £ At A Ti* => y**((A* A B^) s D1.

But by definition of A V B,(AV B)* = A* A Ti*. Since both (A V B)^c(A ® B)**

and 7)^c7)** are von Neumann subalgebras as well as ideals, the quotients are

von Neumann algebras and the induced map

q:(A® B)**/(A V B)^ -> 7)**/7)¿

is also normal. But the first algebra is AV B by definition, while D**/D^

~D**(l-e) by 3.3(v). If k is the normal map k: (A®B)**->AV B, then it

follows from the definition of q that ■ny*if=qk. Thus the above diagram commutes.

The next corollary follows immediately upon combining the diagram for the

C*-tensor product with the previous proposition.

Corollary 4.6. There is a commutative diagram of normal maps with [a, ß] = 0q:

A (A®B)**

Y yf ****■*% Y ^\

A®B -y—¿> < [a>ß] AVB

B

At the expense of omitting the proof of the uniqueness of [a, ß] : A V 7i ->■ D,

the reader can omit the topological considerations (ii) and (iii) in the next lemma

-> (A <g> 7i)** -=-► AV B

D

-y D** -.—► D**(l-e).IT v '



without interrupting the continuity of the rest of the material. Later 4.7(i) is

improved in 5.8.

Lemma 4.7. Consider von Neumann algebras A, 77; their algebraic and von

Neumann tensor products A O 77, A V 77.

(i) There is a natural embedding A O B<=A V 77.

(ii) Furthermore, AO Bis o(A V 77, (A V B)jf)-dense in A V 77.

(iii) Suppose a andß factor through a normal map g: AV B -> D. Then g= [a, ß].

Proof, (i) First it will be shown that for any 0 ̂ t e A O 77, there exists p O q

e Ait O S*, wherep and a are normal, such that </> O q, t}¥=0. Assume A^Jf(H)

and B<îf(K) are von Neumann algebras on Hubert spaces 77 and K. Since

O^teAO 77cjSf(77 ® K), it follows that t(h <g> k)^0 for some heH, k e K.

Thus (t(h (g k), x (g v)t¿0 for some x e 77, y e K. Define p e A* by pa = (ah, x),

and a e 77* similarly as qb = (bk, y). Then

<P Oq,t} = (t(h ®k),x®y)^ 0.

Consequently pOqeA* O B*<=A*AB* and A O Bn(A* A 77*)1 = {0}. Thus

there is an embedding

(ii) Any Banach space £ is o(E**, E*)-dense in 2s**; hence A (g 77 is

ct(04 <g 77)**, (/I <g> 77)*)-dense in (A <g> 77)**. Since A ig 77 was defined as the norm

closure of A O 77, any functional on A (g 77 is completely determined by its value

on A O B. From this it follows that A O 77 is a o((A ® 77)**, (A <g 77)*)-dense subset

of (A® B)**. Now Ait & Bit may be regarded as linear functionals on

(A ® 77)**/C4* A 77*)1 = ^ V 77. Then o(A V B, A* A 77*) is the quotient topology

on A V 77. Thus A O BÂ V 77 remains dense after passage to quotients.

(iii) A C*-homomorphism g: A V77->73 is normal if and only if it is o-

continuous (see [10, p. 1.21, Theorem 5.1]). By the universal property of the C*-

tensor product, if g=g\A (g) 77 is the restriction, then y=g = [a, ß]\A (g 77. Thus

since both [a, ß] and g agree on a a-dense set, [a, ß]=g.

The previous results are summarized in the next theorem and all the topological

parts are isolated out in a corollary.

4.8. Theorem I. Consider von Neumann algebras A = A%, B=B*, their von

Neumann tensor product A V 77, and the product A* A 77* of their preduals. Then

(i) there is a natural embedding of the algebraic tensor product inAOBÂVB;

(ii) the predual ofAVB is A* A 77*.

Suppose a: A -> D and ß: B ->■ D are any normal identity preserving C*-homo-

morphisms with pointwise commuting images into any von Neumann algebra D.


448 J. DAUNS [April

(iii) Then there exists a unique normal C*-homomorphism [a, ß] giving a commuta-

tive diagram

4.9. Corollary to Theorem I. Under the above hypotheses, A O B<=AV B

is a(A V B,(AV B)*)-dense.

5. Functors on C*-aIgebras. The von Neumann tensor product has been

developed with an eye towards constructing a duality theory for locally compact

(nonabelian) groups that would supplement known results (e.g. [5]) in terms of

multiplicative functors, cogebras, and bigebras (e.g. [8] and [13]). Since the in-

dispensable algebra L1(G) does not contain an identity unless the group G is com-

pact, it becomes absolutely necessary to construct a tensor product for C*-algebras

irrespective of whether they do or do not have identity elements. Then it will be

shown that the products eg, V, and A commute with certain natural functors.

Notation 5.1. Consider the category C* of C*-aIgebras and the full sub-

category C*1<=C* of those with identity. The von Neumann algebras, or W*-

algebras, and normal C*-homomorphisms form a subcategory W*^C*\.

Next, four functors E, F, P, and P'1 arc defined. If E: C* -*■ W* is defined by

EA = A**, then let F: W* -> C* be the forgetful functor. For A e W*,setPA = Ajf.

Let 93 be the category of involutive Banach spaces U with U* e W* ; morphisms

are maps/: £/-> F of involutive Banach spaces such that/*: V* -»• U* is in W*.

Thus P is a contravariant functor P: W* —> S3 making W* and 33 dual categories.

For U e 93, let P ~1U= U* e W* be the inverse of P. It should be noted that C*l

and W* contain the subcategories of algebras with identity and identity preserving

maps; there is also a corresponding such subcategory of 93. The above categories

and functors are summarized in the following diagram :

E PC* —^-*■ W* < „ , ' 93

C*l

For the sake of simplicity, notation that by now is standard (see [8] and [13])

will be used without further unnecessary definitions and explanations.

5.2. In order to show that £ is a left adjoint of F, define a set functor

a: W*(E-, _)sC*(_,F-) by

aAM : W*(EA, M) -> C*(A, FM),

n: A**^> M, aAM(n) = n\A,


1972] CATEGORICAL '♦'»-TENSOR PRODUCT 449

where n\A denotes the restriction. For c e C*(A, FM), aA^(c) = c where c factors

through c: A** -> M by the universal property of A**.

The front adjunction -n is a natural transformation of the identity functor 1(C*)

of C* to 77: 1(C*) -> 77Í defined by

^ ■ "a.ea^ba)-^^^**-

In this case it is simply the natural inclusion.

The back adjunction is a functor ifi-.EF^- l(W*) defined as

<Pm = «fm.mÍW): EFM-+M.

For a von Neumann algebra M=M%, in the notation of the previous section,

M$. = M**e and 0 is simply the composition \¡> = (h where M** -> Af**(l— e)

-+ M with M**(l-e)ïM.

The main properties of 17 and i/j will next be described.

5.3. If we only wanted to prove that E is a left adjoint of F, it would simply

suffice to define the natural transformation 77 as above and then show that for any

(notation: V) c e C*(A, FM) there exists a unique (notation: 11) ne W*(EA, M)

such that there is a commutative diagram

FM

Vc e C*(A, FM)

M

3lneW*(EA,M)

The analogous property for the back adjunction is obtained by interchanging the

quantifiers "V" and "3!", interchanging C* and W*, and reversing some arrows:

Vn 6 W*(EA, M) 3 ! c e C*(A, FM)

EA

E(c)

M «-7— EFM FM

5.4. For any algebra A over C, define A1 = A if 1 e A and A1 = CxA otherwise.

As usual, the notation A<\ A1 indicates that algebraically A is an ideal of A1

(not necessarily closed). For nontopological algebras A, B their algebraic tensor

product is defined as A eg B=A1 O B+A O B1.

5.5. For any C*-homomorphism a: A-+B, define another C*-homomorphism

a1: A1 -> B1 by a1 = a if 1 e A, and ^(c, a) = ce+aa if 1 $A and A1 = CxA, where


450 J. DAUNS [April

e is the identity of 771. Note that the restriction of a1 to A is a, but that the kernel

of a1 may be bigger than that of a.

If A and 77 are C*-algebras, it should be recalled that A O 77 denotes the closure

of A O 77 in the norm p (see 2.1) irrespective of whether A and, or, 77 do or do not

contain identities. Since A1 O 77, A O 771 <¡ A1 O B1, and since a sum of closed ideals

is closed in C*-algebras, A1 O 77+^4 O 771 is also a closed ideal in A1 O 771. Define

A®B = AÇ)B1 + A1OB.

For any C*-algebra A (with 1 e A or 1 £ A), define A, = CxA. The categorical

C*-tensor product in the category of all C*-algebras with or without identities,

where homomorphisms need not preserve identities even when they exist, is

A O B, + A, O 77. Suppose a: A-> D, ß:B-^-D are C*-maps with pointwise

commuting images in D. Then a and ß have natural extensions c^: A, -> D„

ß, : B1 -> D,. There is an extension y : A, O B, -»■ T^ and a commutative diagram

Since y(^i O 77) = j8(77) + y04 O 77), it follows that y(A O 2?i + ̂ i O B)*^D^D,.

Now simply define A O B, + A, O 77 -> £> to be the restriction of y to A ® B and

corestriction to D.

5.6. The nonfunctorial nature of the map a -*■ a1 has to be clarified, in order to

show why "_ <g _" is not a functor on the category C*. In general, if a: A -> 77

and 0: 77^ D, then ß1a1==(ß«)1. For example, take .¿^M1, 77=771, and 7J = D1,

but ]8(l)?¿e, where e is the identity of D. Then (ßay(l,0) = e, while (jSVXl, 0)

= (ßa1)(l, 0)=ß(l)#e. Consider the class of all C*-homomorphisms a:A^-B

with the property that if both algebras A and 77 have identities, then a(l) = l.

If 1 £ A or 1 £ 77 or both, a is arbitrary. This class of maps is not closed under

composition. However, if a, ß, and ßa belong to this class, then a straightforward

verification of eight separate cases shows that ß1a1 = (ßa)1.

5.7. Theorem II. For any C*-algebras A, 77 (with or without identities) there

exist monies <f>, p and a commutative diagram where [</>, p] is an isomorphism

E(A ® B)


1972] CATEGORICAL '♦"•-TENSOR PRODUCT 451

Proof. Since AQ> B1<Â® B<=E(A ® 7i) by the universal property of E, there

exists a unique normal map <¡> and a commutative diagram

A->EA

<f>Y , '

AQB1->E(A®B)

Similarly, let p: EB -> E(A <g> 7i). Since A, Ti<= (A Ö B1+A1 Ö B) commute point-

wise, so do also EA, EBÊ(A75 B1+A1Ö B)=E(A® B). Thus <f> and p, define

a unique map [<j>, p]: EA V EB-> E(A ® 7i). By the universal property of JgL,

the functions A-^- EA^>- EAV EB and B-+EAV EB define a unique map

g: A ® B->EA V £2?. By the universal property of 7?, g factors through

g: E(A !§ B)^- EAS7 EB. Thus there is a commutative diagram

EAV EB

// [<f>, P\ \.

EA - -► E(A®B) *-&- EB

\g /Y /

EAV EB *

E(A®B)

Both g [</>, p] and the identity on EA V 2?7i are normal maps. By the uniqueness

for -V-, it follows that g[<f>, p] = 1. By the uniqueness of E, there is at most one

normal map of E(A!§ B)-+ E(A ® 7?) leaving A~® B pointwise fixed. Since

[<f>, p]g is such a map, [</>, p]g= 1. Thus [<f>, p] is an isomorphism whose inverse is g.

The next corollary follows immediately from the fact that E(A ® B)ÊAV EB.

Corollary 5.8. For any C*-algebras A and B, there is a a-dense embedding

A®BcEAVEB.

5.9. Remarks. 1. The previous proposition implies the rather unexpected

result that E(A O B)=£EAV EB in case one of A or 7? does not have an identity.

2. Presently, there is no direct proof of the previous corollary. It does not seem

possible to show directly that (A* V Ti*)1 n A rg" 7i={0} in (A ® 7?)**, or, equiva-

lently, that A* A 7?* separates the points of A eg" 7?.

3. The last proposition also shows that

(A ® 7i)** s (A** ® B**)**/(A* A B*)1.


452 J. DAUNS [April

The conditions 5.10(i)(a), (b) (and (c)) below are the same as those in the defini-

tion of a multiplicative category (with an identity object), see [8, p. 3, Definition

2.1(a) and (b)].

5.10. If A, B and D are H/*-algebras, and C the complex numbers, then

(i)(a) 04 V 77) V D^A V (77 V D);

(b) AV B^BV A;

(c) CVA^A.

(ii) Consequently (W*, V) and (93, A) are multiplicative categories.

Proof, (i) Conclusions (a), (b) and (c) hold, first in the category of rings (with

or without identities) with "_V_" replaced by "-<g_" (see 5.4); and, secondly, for

the category C* of C*-algebras if "_(g-" is used in place of "-V_". Since the

above conclusions hold for either the algebraic or C*-tensor products, and since

either of the latter is o-dense in the IF*-tensor product, (i) follows.

(ii) The restriction and corestriction of a-^-a1 to C*\ is just the identity

functor. Thus _(g_ is just equal to -O— Thus, clearly, _0- is a two variable functor

C*l x C*l -*- C*l. Hence "_V_" is a functor for W* and (W*, V) is a multiplica-

tive category. Since 93 is dual to W*, also (93, A) is a multiplicative category.

The previous results are summarized in the next theorem. As a special case, the

next theorem is applicable to the categories of C*-algebras and H/*-algebras with

identity and identity preserving morphisms.

5.11. Theorem III. Consider the categories C*, W*,and'¡8ofC*-algebras(with

or without identity), von Neumann algebras, and preduals. Let C*1<=C* be the full

subcategory ofiC* containing all C*-algebras with identity. (Note that the morphisms

ofC*\ need not preserve identities.) Let E: C* -> W* be the functor that assigns to a

C*-algebra A its universal von Neumann enveloping algebra EA = A**, while

P: W* -*■ 93 is the contravariant functor where PM is the predual of a W*-algebra

M. Consider the bifunctors ®, V, and A (as defined in 5.6, 4.2, and 2.5) on C*l, W*,

and 93. 77jí>«

(i) (C*l, (g), (W*, V), and (93, A) are multiplicative categories.

(ii) E: (C*l, ig) -+ (W*, V), and P: (W*, V) -> (93, V) are multiplicative func-

tors of multiplicative categories (where P is a duality).

For the next corollary, the reader should be aware that in any one of our three

categories (C*l, ig), (W*, V), or (93, A) the following concepts (see [8] and [13])

are applicable: (1) an algebra over a multiplicative category, (2) a morphism of

algebras (in the sense of (1)), (3) cogebra, (4) morphism of cogebras, (5) bigebra,

(6) morphism of bigebras, and (7) middle four interchange. Also, for a multiplica-

tive category with identity, (l)-(6) have appropriate modifications. In W*, in

addition, all morphisms have to be normal.

5.12. Corollary to Theorem III. The functor E:(C*1,®)^(W*,V) pre-

serves algebras, cogebras, and bigebras. The functor P: (W*, V) -*■ (93, A) trans-

forms



algebras -> cogebras, cogebras -*■ algebras, bigebras -> bigebras.

Parallel statements hold for the categories with identities and identity preserving

morphisms.

6. Examples and applications. A drawback, but at the same time a challenging

aspect of the present tensor product -V-, is that it can very rarely be computed

in a rigorous way in concrete examples, unless one is willing to make some

assumptions.

First, the new tensor product -V_ is compared with the usual one.

6.1. Suppose A = A"^^C(H) and B=B"^if(K) are von Neumann algebras.

Frequently, the double commutant (A O B)"^¿f(H® K) of the algebraic tensor

product is referred to as the von Neumann tensor product. Denote the operator

norm closure of A O B by A D B. Let A* D 7?*c,4* D 7i*c=(^ □ 7?)* denote the

closure of the algebraic tensor products in the dual norm of the operator norm on

SC(H® K). As before, (A* D B*)l<=(A D B)** is the annihilator of the transla-

tion invariant norm closed subspace A* □ 7?*. In [10, p. 3.17, Theorem 2.3] it is

shown that (A O B)"^(A □ B)**/(A* D 7?*)1. By the universal property of _V_,

there is a normal map X: AS7 B^*(A O B)". The image A(^ V 7?) of a von Neu-

mann algebra AV7 B under a normal homomorphism A is a von Neumann sub-

algebra of (A O B)". Bufr/Í O Tic X(A OB)Q(AO B)", and (A O B)" is the smallest

von Neumann subalgebra containing AQ B. Therefore A(,4 V 7?) = (A O B)".

Next, it is shown that A V B contains a copy of A □ B.

6.2. For any normal map A of any two von Neumann algebras, A V B=K® P,

where K is the kernel of A, and where K and P are von Neumann ideals of A V B.

I.e. arbitrary least upper bounds of sets of positive elements in Tv, F agree with those

in A V B. Then P^(A O B)". Let J= X~\A U B). Since K^J, J=K® (J n P).

Also A O 7i<=7 and J n Fs A □ B. Since A is the identity on A O B, A O B n K={0}

and the projection p: A O B -^- J r\ P is one-to-one. A copy of A □ B may be found

inside A V 7? as follows :

A O B ^ p(A O B) ^ J n P ^ AU B.

Note that A Q B does not contain the canonical copy of A O Ti.

It will be shown that X(A O B) = A □ T?. Since A O B<=J, since 7 is norm closed,

and since the closure of A O B in A V 7? is A Ö 7?, it follows that A Ö B=J- Thus

AO B<^X(AOB)^AU B. However, X(A Ö B) is norm closed in A □ 7?, while

^ □ 2? is the norm closure of A O B. Thus X(A~Ö B) = AU B.

The next example shows that the usual tensor product of von Neumann algebras

and _V_ are different.

6.3. Example. Consider a type II factor A cj£?(77). The commutant B=A' of A

is also a type II factor. It suffices to show that (A O B)" does not have the universal

property. It follows from [10, III, p. 3.40, Theorem 4.3] that also (A Q B)" is of

type II. Also, (A Q B)" is a factor. If a and ß are the natural inclusions, suppose


454 J. DAUNS [April

that the following commutative diagram can be completed by a normal homo-

morphism g

The image of any W*-a\gebra under a normal map is a von Neumann subalgebra.

Then g((A O 77)") is a von Neumann subalgebra of if(H). However, A, B

<^g((AO 77)") and the von Neumann subalgebra generated by A and 77is all of if(H).

Thus g HA O B)") = if(H). However, if(H) is of type I, while g((A O 77)") is of

type II, a contradiction. Thus there cannot exist such a map g.

Since A V 77 £ (A O 77)", the map À: A V 77 -> (A Q 77)" of 6.1 must have a non-

trivial kernel. If A and 77 are type II finite factors, then they are algebraically simple.

Also (A O 77)" is known to be algebraically simple. Thus this same example also

serves to illustrate that even though A and 77 are algebraically simple, A V 77 need

not be.

6.4. Any commutative von Neumann algebra M is of the form M=Lco(X) on

some measure space with Mif=L1(X). Consider a second commutative von

Neumann algebra N=Lco(Y). Denote the least and greatest cross norms, as usual,

by A and y. The norm p used to form M (g N is A. Thus the norm on M* ® TV*

is p' = X'=y and M* ® N*=L\X) ® L\Y)^L\Xx Y). In general, the inclusion

M®N^Lx(XxY) is proper. Thus (M® N)*^Lx(Xx Y)*/(M® N)L where

L™(Xx Y)* is the vector space of regular Borel measures on the maximal ideal

space of L'°(Xx Y).

6.5. Assume Af* ® 7V* = M* A TV*. Then M V TV=(A7* ® N*)*=Lco(Xx Y).

Thus M ig TV/Af V TV. Here M V TV=(M O TV)" if LîX^&iLîX)) acts by

multiplication. As Hubert spaces, Z,2^) <g L2( f)?L2(Ix 7). It is easily seen that

(L°°(Ar) o Lœ(F)y = (Lœor) o ¿"(r))" = r(ix y).

6.6. If X= Y= G is a locally compact group with the left invariant measure, then,

as is well known, LCC(G) is a cogebra in (W*, V) under the comultiplication

c: LX(G) ->LX(G x G), where (cf)(y, z)=f(yz) for y,zeG. Then P transforms it

into an ordinary Banach algebra P(LX(G))=L1(G) in 93 where the multiplication

is the usual convolution.

The objective of the next example is to verify directly some of the previous results

by a direct computation.

Example 6.7. Consider a Hubert space 77, the algebra if of all bounded opera-

tors, and the three ideals—the trace-class, Schmidt-class, and compact operators

¡rîfîf^îf respectively. If {e(i)}<^H is an arbitrary orthonormal basis then

recall that P e if if and only if 2 ||Pe(0 II2 < °o and that P= if2={PQ \P,Qe if}.


1972] CATEGORICAL ^»-TENSOR PRODUCT 455

For Te3T, the trace tr (T) of T is tr (T) = Z(Te(i), e(i)). Then if(€*=^ and

F* = if, where for N e if and T e 3T, (J, 7Y> = tr (7VT).

Suppose 7> = .S? = 7>*, where 7>* = ¿T. Then D* = D*@eD*, where 7)^ = 7)**«?

<=7)**. Every element fe D* is uniquely of the form/= 7*+8 with SeeD* and

Te3~ where F is the restriction T=f\Sfc€. Since ?>\if(êeifc€* = 3~, and since

^" n e2)* = {0}, it is reasonable to suppose and indeed can be shown [12, p. 50,

Theorem 5] that <8, jSfr> = {0}. Thus the singular part eD* of D* vanishes on Sf<€.

Then 7>** = 7>i © 7>**(1 -e). By 3.3(iii) and (v), 7>**(1 -e)^D% = D. Here the

latter conclusion may be verified directly, since 7)*=^" and we know that 9~*

= Sf = D.

Let us interpret the conclusion in 3.6 that r¡D n D**e = {0}. Suppose Be if

with r,B e D**e. But by definition of e, D**e = D$¡=S>~1. Thus for all Te ST,

<t,B, F> = <T, 7?> = tr (BT) = 0.

But it is shown in [12, p. 45, Lemma 1] that tr (BX)=0 for all operators leJ

of rank one if and only if B=0. Thus we have verified directly that r¡D n D**e = {0}.

The objective of the next example is to illustrate the conclusion 77^ n A**e = {0}

in Proposition 3.5.

Example 6.8. Consider A = if(H) for an infinite dimensional Hilbert space 77.

Every functional/e A*—the predual of A—is obtained from two square summable

sequences of vectors {x(n)}, {y(n)}^H by f(T) = J, (Tx(n), y(n)) for Te A (see [10,

III, p. 3.7, Proposition 1.1] or [20, p. 21, Lemma 6]). If -qA^A**, then 77(F): A*

-> C and <[r¡T,fy=f(T). If O^Te A is arbitrary, then there is an /as above, in

fact of the simple form f(T) = (Tx,y), with/(F) ^0. Thus r¡(T)i A^ = A**e and

77^ nA**e = {0}.

6.9. In conclusion, some unsolved problems are listed.

(1) Perhaps an intrinsic tensor product can be defined on all complex Banach

spaces such that, when restricted to 93, it agrees with _A_.

(2) In the notation of 3.6, can the intersection 77^ n A**(I —e) be characterized?

(3) Under what conditions do the two tensor products agree :

{AOB) - (A^ □ B*y - (At A B*y = AVB-

(4) To find examples with /1*®2?*#/Í*A 2?*.

(5) Does At O 7?* separate the points of A ® 2??

References

1. J. Dauns and K. H. Hofmann, Representation of rings by sections, Mem. Amer. Math.

Soc. No. 83, Amer. Math. Soc, Providence, R. I., 1968, pp. 1-180. MR 40 #752.

2. -, Spectral theory of algebras and adjunction of identity, Math. Ann. 179 (1969),

175-202. MR 40 #734.

3. J. Dauns, Multiplier rings and primitive ideals, Trans. Amer. Math. Soc. 145 (1969),

125-158. MR 41 #1805.


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4. J. Dixmier, Les C*-algèbres et leurs représentations, 2nd ed., Cahiers Scientifiques, fase.

29, Gauthier-Villars, Paris, 1969. MR 39 #7442.

5. P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France

92 (1964), 181-236. MR 37 #4208.

6. A. Guichardet, Tensor products of C*-algebras. Part I: Finite tensor products, Lecture

Note Series, no. 12, Matematisk Institut, Aarhus Universitet, 1969.

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Series, no. 13, Matematisk Institut, Aarhus Universitet, 1969.

8. K. H. Hofmann, The duality of compact semigroups and C*-bialgebras, Lecture Notes in

Math., vol. 129, Springer-Verlag, Berlin-New York, 1970, pp. 1-141.

9. Y. Misonou, On the direct product of W*-algebras, Tôhoku Math. J. (2) 6 (1954), 189-204.

MR 16, 1125.

10. S. Sakai, The theory of W*-algebras, Lecture Notes, Yale University, New Haven, Conn.,

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Press, Princeton, N. J., 1950. MR 12, 186.

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ihrer Grenzgebiete, N.F., Heft 27, Springer-Verlag, Berlin, 1960, pp. 1-81. MR 22 #9878.

13. E. M. Sweedler, Hopf algebras, Math. Lecture Note Series, Benjamin, New York, 1969.

MR 40 #5705.

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299-304. MR 16, 146.

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212-219. MR 16, 1126.

16. -, Conjugate spaces of operator algebras, Proc. Japan Acad. 30 (1954), 90-95.

MR 16, 146.17. M. Takesaki, On the conjugate space of operator algebra, Tôhoku Math. J. (2) 10 (1958),

194-203. MR 20 #7227.18. -, On the cross-norm of the direct product of C*-algebras, Tôhoku Math. J. (2) 16

(1964), 111-122. MR 29 #2668.19. T. Turumaru, On the direct-product of operator algebras. I, Tôhoku Math. J. (2) 4 (1952),

242-251. MR 14,991.20. A. Wulfsohn, Produit tensoriel de C*-algèbres, Bull. Sei. Math. (2) 87 (1963), 13-27.

MR 29 #5121.

Department of Mathematics, Tulane University, New Orleans, Louisiana 70118


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